View source for Baire classes

Families of real functions which are defined inductively using the ordinal number of limit transitions involved in the definition of the function, and which constitute the classification of functions proposed in 1899 by R. Baire [[#References|[1]]], known as the Baire classification. The set of all continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150302.png" /> is a metric space, is called the zero-th Baire class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150303.png" />. The first Baire class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150304.png" /> is the set of discontinuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150305.png" /> which are the limit of a sequence of continuous functions converging at each point. The Baire class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150307.png" /> is an ordinal number of the first or second class, is defined as the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150308.png" /> which do not form part of any of the preceding classes, but which may be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b0150309.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503011.png" />. The union of the Baire classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503012.png" /> over all ordinal numbers of the first and second class constitutes the set of Baire functions. This is the minimal class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503013.png" /> containing all the continuous functions and closed with respect to pointwise convergence. A linear combination, product and quotient (for non-zero denominators) of functions of a Baire class not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503014.png" /> is a function in a Baire class not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503015.png" />. A uniformly convergent sequence of functions of a Baire class not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503016.png" /> has as limit a function in a Baire class not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503017.png" />. Necessary and sufficient conditions have been established for a sequence of functions in a Baire class not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503018.png" /> to converge to a function in a Baire class not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503019.png" /> [[#References|[4]]]. The union of all dense-in-itself sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503020.png" /> is called the kernel of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503022.png" /> is a complete space with a non-empty dense-in-itself kernel, none of the Baire classes is empty [[#References|[2]]]. The set of Baire functions coincides with the set of Borel-measurable functions (cf. [[Borel measure|Borel measure]]), and for this reason they are all Lebesgue-measurable (cf. [[Lebesgue measure|Lebesgue measure]]). A Lebesgue-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503023.png" /> is equivalent to a Baire function of a Baire class not higher than two [[#References|[3]]]. Baire, who considered functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503024.png" /> (mainly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503025.png" />), made the most detailed study of functions of the first class. He showed that a necessary and sufficient condition for a discontinuous function to belong to the first class is the existence of a point of continuity of the induced function on each perfect set (Baire's theorem). This theorem is applicable to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503027.png" /> has the [[Baire property|Baire property]] [[#References|[2]]]. The concept of a Baire function can be naturally generalized to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503029.png" /> is an arbitrary metric space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Baire,   "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,   "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,   "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.M. Gagaeff,   "Sur les suites convergentes de fonctions mesurables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015030/b01503030.png" />"  ''Fund. Math.'' , '''18'''  (1932)  pp. 182–188</TD></TR></table>



====Comments====
A subset of a topological space is called dense-in-itself if it has no relatively-isolated points. A modern English reference for the notion of Baire classes is [[#References|[a1]]]. For an introduction to the topic and some applications, cf. [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C.M. van Rooy,   W.H. Schikhof,   "A second course on real functions" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.P Boas jr.,   "A primer of real functions" , Math. Assoc. Amer.  (1981)</TD></TR></table>